Move Animation within the Timeline
Moving a Keyframe animation in time:
You can move the entire animation sequence from within the Dope Sheet Editor. There you select all keyframes (type "a" on the keyboard once or twice), then for example: G X 30  to shift the animation by 30 frames:


Moving a Constrained Path Animation in time:
This animation type starts at timeframe 0 by default. Within the Constraints panel you can define an Offset to shift the start of the animation:


Open The Object Constraints of the animated Object
Set the Offset to the start timeframe where the Path Animation shall begin (30 in this case) 



Moving the Animation time of an Object parented to a Path

This animation type starts at timeframe 0 by default and runs for 100 Frames, then it stops. You can define the animation window by setting 2 keyframes in the Path Animation Panel of the Path:


Select the Path
In the Data Properties locate the Path Animation panel
In the Timeline select the startframe for the animation (30 in this case)
In the Path Animation panel set the Evaluation time to 0.00
Then keyframe the Evaluation time.
In the timeline select the Endframe for the Animation
In the Path Panel set the Evaluation time to 100 (or to whatever value you have set the Frames Attribute right above the Evaluation tmie attribute)
Keyframe the Evaluation time again.